# Quantum measurements with prescribed symmetry

## TL;DR

This paper presents a method to identify whether quantum measurements are unique or part of a symmetric family, with applications to SIC-POVMs, mutually unbiased bases, and Kochen-Specker sets across various dimensions.

## Contribution

It introduces a linear-system-based technique to analyze the symmetry properties of quantum measurements and applies it to derive maximal symmetric measurement sets in multiple dimensions.

## Key findings

- Maximal SIC-POVM in dimension 3 derived
- Certain measurement sets are shown to be isolated with no free parameters
- The method applies to mutually unbiased bases and Kochen-Specker sets in specified dimensions

## Abstract

We introduce a method to determine whether a given generalised quantum measurement is isolated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of equations in some relevant cases. As consequence, we provide a simple derivation of the maximal family of Symmetric Informationally Complete measurements (SIC)-POVM in dimension 3. Furthermore, we show that the following remarkable geometrical structures are isolated, so that free parameters cannot be introduced: (a) maximal sets of mutually unbiased bases in prime power dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c) contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18 and 21 vectors, respectively.